Problem 32 - College Algebra

Question

In Exercises \(29-36,\) find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. \(n=3 ; 6\) and \(-5+2 i\) are zeros; \(f(2)=-636\)

Step-by-Step Solution

Answer

The polynomial function is \(f(x) = -3x^3 - 12x^2 -117x + 90\).

Step 1: Identify the Roots

Given \(n=3\) and two roots \(6\) and \(-5+2i\). Since polynomials with real coefficients always have complex roots in conjugate pairs, the third root must be the conjugate of the complex root, \(-5-2i\).

Step 2: Construct the Polynomial

The polynomial can be formed by using the roots in the form of the binomial theorem, \((x - a)(x - b)(x - c)\), where \(a, b,\) and \(c\) are the roots. Written out, the polynomial takes the form: \((x - 6)(x - (-5 + 2i))(x - (-5 - 2i))\)

Step 3: Multiply the Binomials

When we multiply the binomials together, we get the polynomial: \((x - 6)(x + 5 - 2i)(x + 5 + 2i)\). Expanding this: \(x^3 + 4x^2 +39x - 30\)

Step 4: Calculate the Leading Coefficient

We can calculate the leading coefficient by using the condition \(f(2) = -636\). This gives us the equation \(a(2)^3 + 4a(2)^2 +39a(2) - 30a = -636\). Solving for \(a\), we find that \(a= -3\)

Step 5: Write Polynomial

Now, substitute \(a = -3\) back into the polynomial \(-3x^3 - 12x^2 -117x + 90\)

Key Concepts

Understanding Complex Roots

Complex roots are solutions to polynomial equations that include imaginary numbers. Imaginary numbers are expressed with the letter "i," which is the square root of -1. In the example exercise, -5 + 2i is a complex root. In mathematics, when dealing with polynomials that have real coefficients, complex roots always come in pairs. These pairs are called "conjugates," which means that if one root is -5 + 2i, its conjugate pair is -5 - 2i.

• Complex roots appear in pairs for real-coefficient polynomials.
• The conjugate of -5 + 2i is -5 - 2i.

Without pairing them as conjugates, the polynomial wouldn't simplify to have real-number coefficients. This characteristic is essential when constructing polynomial functions with real numbers.

Real Coefficients in Polynomials

In mathematics, the term "real coefficients" means that the numbers in front of the variables within the polynomial are real numbers, not imaginary or complex. When you compose a polynomial with real coefficients, like in our exercise, it's important to remember that roots that are complex must have their conjugates included in the polynomial to maintain real coefficients across all terms.

• Real coefficients make the polynomial applicable to real-world situations.
• Ensure that each complex root has a conjugate to keep coefficients real.

Polynomials with real coefficients reflect natural patterns and phenomena better than their complex or exclusively imaginary counterparts. This makes them particularly useful in practical engineering and physics applications.

Constructing a Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In our original exercise, the goal is to construct an nth-degree polynomial. Given roots such as 6, -5 + 2i, and -5 - 2i (the conjugate of -5 + 2i), these roots form the foundation.
By setting the polynomial with these roots, you use the binomial theorem's format: \[ (x - 6)(x + 5 - 2i)(x + 5 + 2i) \]Multiplying these binomials allows us to expand and form the complete polynomial. If conditions specify that the function passes through a certain point, like \( f(2) = -636 \), further calculations refine the leading coefficient to match. In our exercise, the polynomial therefore transforms into:\[ -3x^3 - 12x^2 - 117x + 90 \]

• Start with known roots, multiplying binomials guides the expansion.
• Adjust the leading coefficient to adhere to given conditions like function values.

Such methods of constructing polynomial functions are pivotal in both theoretical mathematics and practical applications across various scientific disciplines.